3.648 \(\int \frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=259 \[ -a^{3/2} \sqrt{c} (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 d}-\frac{\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{3/2}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac{4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac{b \sqrt{a+b x} (c+d x)^{3/2} (7 a d+b c)}{4 d} \]

[Out]

-((b^2*c^2 - 14*a*b*c*d - 19*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d) + (b*(b
*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(4*d) + (4*b*(a + b*x)^(3/2)*(c + d*x
)^(3/2))/3 - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/x - a^(3/2)*Sqrt[c]*(5*b*c + 3*a*
d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b^3*c^3 - 15*a*b
^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(8*Sqrt[b]*d^(3/2))

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Rubi [A]  time = 0.916149, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -a^{3/2} \sqrt{c} (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 d}-\frac{\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{3/2}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac{4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac{b \sqrt{a+b x} (c+d x)^{3/2} (7 a d+b c)}{4 d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

-((b^2*c^2 - 14*a*b*c*d - 19*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d) + (b*(b
*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(4*d) + (4*b*(a + b*x)^(3/2)*(c + d*x
)^(3/2))/3 - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/x - a^(3/2)*Sqrt[c]*(5*b*c + 3*a*
d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b^3*c^3 - 15*a*b
^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(8*Sqrt[b]*d^(3/2))

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Rubi in Sympy [A]  time = 121.802, size = 241, normalized size = 0.93 \[ - a^{\frac{3}{2}} \sqrt{c} \left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{4 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3} + \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (\frac{7 a d}{4} + \frac{b c}{4}\right ) - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (5 a^{2} d^{2} + 26 a b c d + b^{2} c^{2}\right )}{8 d} + \frac{\left (5 a^{3} d^{3} + 45 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{8 \sqrt{b} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**2,x)

[Out]

-a**(3/2)*sqrt(c)*(3*a*d + 5*b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c +
d*x))) + 4*b*(a + b*x)**(3/2)*(c + d*x)**(3/2)/3 + (a + b*x)**(3/2)*sqrt(c + d*x
)*(7*a*d/4 + b*c/4) - (a + b*x)**(5/2)*(c + d*x)**(3/2)/x + sqrt(a + b*x)*sqrt(c
+ d*x)*(5*a**2*d**2 + 26*a*b*c*d + b**2*c**2)/(8*d) + (5*a**3*d**3 + 45*a**2*b*
c*d**2 + 15*a*b**2*c**2*d - b**3*c**3)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt
(a + b*x)))/(8*sqrt(b)*d**(3/2))

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Mathematica [A]  time = 0.25124, size = 265, normalized size = 1.02 \[ \frac{1}{16} \left (8 a^{3/2} \sqrt{c} \log (x) (3 a d+5 b c)-8 a^{3/2} \sqrt{c} (3 a d+5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d (11 d x-8 c)+2 a b d x (34 c+13 d x)+b^2 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{3 d x}+\frac{\left (5 a^3 d^3+45 a^2 b c d^2+15 a b^2 c^2 d-b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} d^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

((2*Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d*(-8*c + 11*d*x) + 2*a*b*d*x*(34*c + 13*
d*x) + b^2*x*(3*c^2 + 14*c*d*x + 8*d^2*x^2)))/(3*d*x) + 8*a^(3/2)*Sqrt[c]*(5*b*c
+ 3*a*d)*Log[x] - 8*a^(3/2)*Sqrt[c]*(5*b*c + 3*a*d)*Log[2*a*c + b*c*x + a*d*x +
2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]] + ((-(b^3*c^3) + 15*a*b^2*c^2*d
+ 45*a^2*b*c*d^2 + 5*a^3*d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/(Sqrt[b]*d^(3/2)))/16

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Maple [B]  time = 0.024, size = 696, normalized size = 2.7 \[{\frac{1}{48\,dx}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{3}{b}^{2}{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+15\,{d}^{3}{a}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x\sqrt{ac}+135\,{d}^{2}{a}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cbx\sqrt{ac}+45\,da{b}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}x\sqrt{ac}-3\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{3}x\sqrt{ac}-72\,{a}^{3}c\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){d}^{2}\sqrt{bd}x-120\,{a}^{2}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) db\sqrt{bd}x+52\,{x}^{2}a{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}\sqrt{ac}+28\,{x}^{2}{b}^{2}cd\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+66\,{d}^{2}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}x\sqrt{ac}+136\,acd\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}x\sqrt{ac}+6\,{b}^{2}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}x\sqrt{ac}-48\,{a}^{2}cd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^2,x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*x^3*b^2*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)+15*d^3*a^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*(a*c)^(1/2)+135*d^2*a^2*ln(1/2*(2*b*d*
x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c*b*x*(a*c
)^(1/2)+45*d*a*b^2*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)
+a*d+b*c)/(b*d)^(1/2))*c^2*x*(a*c)^(1/2)-3*b^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^3*x*(a*c)^(1/2)-72*a^3*c*ln
((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*d^2*(b*d)^
(1/2)*x-120*a^2*c^2*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)+2*a*c)/x)*d*b*(b*d)^(1/2)*x+52*x^2*a*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*
d)^(1/2)*(a*c)^(1/2)+28*x^2*b^2*c*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*
(a*c)^(1/2)+66*d^2*a^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*x*(a*c)^(1/2)
+136*a*c*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*x*(a*c)^(1/2)+6*b^2*c^2
*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*x*(a*c)^(1/2)-48*a^2*c*d*(a*c)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/d
/(b*d)^(1/2)/(a*c)^(1/2)/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.03305, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^2,x, algorithm="fricas")

[Out]

[1/96*(24*(5*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sqrt(b*d)*x*log((8*a^2*c^2 + (b^2*c^
2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)
*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 3*(b^3*c^3 - 15*a*b^2*c^2*d - 4
5*a^2*b*c*d^2 - 5*a^3*d^3)*x*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x +
a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d +
a*b*d^2)*x)*sqrt(b*d)) + 4*(8*b^2*d^2*x^3 - 24*a^2*c*d + 2*(7*b^2*c*d + 13*a*b*
d^2)*x^2 + (3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt
(d*x + c))/(sqrt(b*d)*d*x), 1/48*(12*(5*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sqrt(-b*d
)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d
)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 3*(
b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*x*arctan(1/2*(2*b*d*x + b
*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) + 2*(8*b^2*d^2*x^3 - 24*
a^2*c*d + 2*(7*b^2*c*d + 13*a*b*d^2)*x^2 + (3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)
*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*d*x), -1/96*(48*(5*a*b*c
*d + 3*a^2*d^2)*sqrt(-a*c)*sqrt(b*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(
-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) + 3*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d
^2 - 5*a^3*d^3)*x*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x
+ c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x
)*sqrt(b*d)) - 4*(8*b^2*d^2*x^3 - 24*a^2*c*d + 2*(7*b^2*c*d + 13*a*b*d^2)*x^2 +
(3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/
(sqrt(b*d)*d*x), -1/48*(24*(5*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*sqrt(-b*d)*x*arcta
n(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) + 3*(b^3
*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*x*arctan(1/2*(2*b*d*x + b*c
+ a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) - 2*(8*b^2*d^2*x^3 - 24*a^2
*c*d + 2*(7*b^2*c*d + 13*a*b*d^2)*x^2 + (3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)*x)
*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*d*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.655033, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^2,x, algorithm="giac")

[Out]

sage0*x